Steve Simpson asks whether (and thereby insinuates that) I am
"ambiguating" on the distinction between second order logic with Henkin
semantics and second order logic with standard semantics. When I say
that the very same axioms and rules that are complete for Henkin
semantics are sound for standard semantics, I mean by "standard
semantics" the semantics for second order languages in accordance with
which the monadic predicate variables are to be construed as ranging
over the power set of the domain, S, of the structure in question, the
binary predicate variable are to be construed as ranging over the power
set of (S X S), etc.
Now it's pretty rich of Simpson to insinuate that I am
"ambiguating" when he goes on to ambiguate away the notion of standard
semantics itself. I feel like a man who has been accused of insobriety
by the town drunk.
I say that the very formal axioms and rules that are complete
for Henkin semantics are sound for standard semantics. "Not so." says
Simpson "It depend on the meta-theory." Now what possible meaning could
we attach to Simpson's words other than that the notion of standard
semantics for second order logic is, well, ambiguous? But what is the
nature of this ambiguity? Is it that the meaning of "standard 2nd order
semantics" varies from model to model of 1st order ZF? Or is it that
what we can formally prove about standard second order semantics
depends upon which consistent extension of 1st order ZF we choose as
our meta-theory. If it is the latter, then Simpson stands exposed as a
man whose *instincts* are those of an old fashioned formalist, even if
he expressly denies being one. But if it the former possibility, if
it's a matter of the meaning of "standard semantics" varying from model
to model of 1st order ZF, then there is a really piquant irony in
Simpson's position.
Since even the notion of "natural number" is ambiguous if we
move from model to model 1st order ZF, we must surely confine ourselves
to some restricted class of "relevant" models. What restrictions should
we impose? Well- foundedness? That will at least get rid of the
ambiguity concerning natural numbers. But then by Mostowski's Lemma, we
can restrict ourselves further to *standard* models in which the
universe of discourse consists entirely of sets and the epsilon symbol
is interpreted as set membership. But wait a minute: there are
countable standard models in which, necessarily, the notion of "real
number" is ambiguous, and in which "power set" doesn't really mean
"power set". So maybe we ought to consider only models in which "power
set" is absolute. There's still a difficulty though: some of those
models are omega cofinal. Do we really want to include *them*? Or
models which are alpha-cofinal for some alpha actually lying in the
model? But where have we now arrived? The class of models we are
interested in are precisely the models of . . . *second order ZF*. Do
we see the problem of the Lowenheim number of second order logic
looming on our horizon? How fortunate that Simpson has recently become
aware that there might be such a thing as this number.
Let me make it clear that I am not suggesting that we take
second order ZF (with some system axioms and rules for second order
logic) as our base formal theory for investigating provability in set
theory: first order ZF is OBVIOUSLY a much more sensible choice, as I'm
sure we all agree. And one other thing: I have been teasing Steve
Simpson about being an "old fashioned" formalist. But it is clear to me
that being a formalist, "old fashioned" or not, is not going to prevent
you from doing first rate work in mathematical logic. (Simpson's own
case - if he is a formalist - proves that.) Nor is it going to prevent
you from doing first rate work in axiomatic set theory - I think maybe
even Paul Cohen was a formalist when he did his epoch-making work. But
it *is* going to prevent you from giving a coherent account of the
foundations of mathematics.
John Mayberry
School of Mathematics
University of Bristol
-----------------------------
John Mayberry
J.P.Mayberry at bristol.ac.uk
-----------------------------